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Critical Point (mathematics), Critical Point
Critical point may refer to: *Critical phenomena in physics *Critical point (mathematics), in calculus, a point where a function's derivative is either zero or nonexistent *Critical point (set theory), an elementary embedding of a transitive class into another transitive class which is the smallest ordinal which is not mapped to itself *Critical point (thermodynamics), a temperature and pressure of a material beyond which there is no longer any difference between the liquid and gas phases *Quantum critical point *Critical point (network science) *Construction point, in skiing, a line that represents the steepest point on a hill See also *Critical value (other) *Critical path (other) *Brillouin zone *Percolation thresholds {{DEFAULTSORT:Critical Point Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Phenomena
In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such as the magnetic susceptibility in the ferromagnetic phase transition) described by critical exponents, universality, fractal behaviour, and ergodicity breaking. Critical phenomena take place in second order phase transitions, although not exclusively. The critical behavior is usually different from the mean-field approximation which is valid away from the phase transition, since the latter neglects correlations, which become increasingly important as the system approaches the critical point where the correlation length diverges. Many properties of the critical behavior of a system can be derived in the framework of the renormalization group. In order to expl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (set Theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself. p. 323 Suppose that j: N \to M is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(\omega) = \omega. If j(\alpha) = \alpha for all \alpha \kappa, then \kappa is said to be the critical point of j. If N is '' V'', then \kappa (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number ''κ'' such that there exists a \kappa-complete, non-principal ultrafilter over \kappa. Specifically, one may take the filter to be \. Generally, there will be many other <''κ''-complete, non-principal ultrafilters over . However, might be different from the |
Critical Point (thermodynamics)
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a ''critical temperature'' ''T''c and a ''critical pressure'' ''p''c, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition (Curie temperature) in the absence of an external magnetic field. Liquid–vapor critical point Overview For simplicity and clarity, the generic notion of ''critical point'' is best introduced by discussing a specific example, the vapor–liquid critical point. This was the first critical point to be discovered, and it is still the best known and most studied one. The figu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Critical Point
A quantum critical point is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero. A quantum critical point is typically achieved by a continuous suppression of a nonzero temperature phase transition to zero temperature by the application of a pressure, field, or through doping. Conventional phase transitions occur at nonzero temperature when the growth of random thermal fluctuations leads to a change in the physical state of a system. Condensed matter physics research over the past few decades has revealed a new class of phase transitions called quantum phase transitions which take place at absolute zero. In the absence of the thermal fluctuations which trigger conventional phase transitions, quantum phase transitions are driven by the zero point quantum fluctuations associated with Heisenberg's uncertainty principle. Overview Within the class of phase transitions, there are two main categories: at a ''first-order phase t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (network Science)
In network science, a critical point is a value of average degree, which separates random networks that have a giant component from those that do not (i.e. it separates a network in a subcritical regime from one in a supercritical regime). Considering a random network with an average degree \langle k\rangle the critical point is \langle k\rangle = 1 where the average degree is defined by the fraction of the number of edges (e) and nodes (N) in the network, that is \langle k\rangle =\frac. Subcritical regime In a subcritical regime the network has no giant component, only small clusters. In the special case of \langle k\rangle =0 the network is not connected at all. A random network is in a subcritical regime until the average degree exceeds the critical point, that is the network is in a subcritical regime as long as \langle k\rangle 1. Example on different regimes Consider a speed dating event as an example, with the participants as the nodes of the network. At the begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Construction Point
The construction point ( ger, Konstruktionspunkt), also known as the K-point or K-spot and formerly critical point, is a line across a ski jumping hill. It is used to calculate the number of points granted for a given jump. It is therefore also called calculation point or calculation line. Classification The distance between construction point and take-off table, measured in meters, was formerly used to classify the size of a ski jumping hill. Since mid-2004, the hills are instead categorized by the hill size. Nearly all competitions in the FIS Ski Jumping World Cup use large hills with a construction point between 120 and 130. The largest is Mühlenkopfschanze in Germany. In addition, there is a bi-annual FIS Ski-Flying World Championship, which is held in one of the world's five ski flying hills: Vikersundbakken in Norway, Letalnica Bratov Gorišek in Slovenia, Čerťák in the Czech Republic, Heini Klopfer Ski Jump in Germany and Kulm in Austria. In the FIS Ski Jumping Conti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Value (other)
Critical value may refer to: *In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''. *In statistical hypothesis testing, the critical values of a statistical test are the boundaries of the acceptance region of the test. The acceptance region is the set of values of the test statistic for which the null hypothesis is not rejected. Depending on the shape of the acceptance region, there can be one or more than one critical value. *In complex dynamics, a critical value Critical value may refer to: *In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''. *In statistical hypothesis ... is the image of a critical point. *In medicine, a critical value or panic value is a value of a laboratory test that indicates a serio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Path (other)
Critical path may refer to: *The longest series of sequential operations in a parallel computation; see analysis of parallel algorithms * Critical path method, an algorithm for scheduling a set of project activities *Critical path drag, a project management metric *Critical path network diagram, a network diagram highlighting the critical path * ''Critical Path'' (book), by Buckminster Fuller *''The Critical Path: An Essay on the Social Context of Literary Criticism'', a 1971 book by Northrop Frye *''The Critical Path'', a podcast by Horace Dediu * ''Critical Path'' (video game), an interactive movie computer game * Critical Path, Inc., a provider of messaging services *Critical Path Institute, an organization for improvement of the drug development process *Critical Path Project, a video archive *Critical Path Project, early source of HIV/AIDS information founded by Kiyoshi Kuromiya See also *Critical graph, a graph where every vertex or edge is a critical element *Critical mass (di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in ''k''-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Vor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Thresholds
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability ''p'', or more generally a critical surface for a group of parameters ''p''1, ''p''2, ..., such that infinite connectivity (''percolation'') first occurs. Percolation models The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites (vertices) or bonds (edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
